Combinatorial Control through Allostery
CCombinatorial Control through Allostery
Vahe Galstyan, † , ⊥ Luke Funk, ‡ , ⊥ Tal Einav, ¶ and Rob Phillips ∗ , ¶ , § , (cid:107) † Biochemistry and Molecular Biophysics Option, California Institute of Technology,Pasadena, California 91125, United States ‡ Harvard-MIT Division of Health Sciences and Technology and the Broad Institute of MITand Harvard, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139,United States ¶ Department of Physics, California Institute of Technology, Pasadena, California 91125,United States § Department of Applied Physics, California Institute of Technology, Pasadena, California91125, United States (cid:107)
Division of Biology and Biological Engineering, California Institute of Technology,Pasadena, California 91125, United States ⊥ Contributed equally to this work
E-mail: [email protected]
Phone: (626) 395-33741 a r X i v : . [ q - b i o . B M ] D ec bstract Many instances of cellular signaling and transcriptional regulation involve switch-like molecular responses to the presence or absence of input ligands. To understandhow these responses come about and how they can be harnessed, we develop a statis-tical mechanical model to characterize the types of Boolean logic that can arise fromallosteric molecules following the Monod-Wyman-Changeux (MWC) model. Buildingupon previous work, we show how an allosteric molecule regulated by two inputs canelicit AND, OR, NAND and NOR responses, but is unable to realize XOR or XNORgates. Next, we demonstrate the ability of an MWC molecule to perform ratiometricsensing - a response behavior where activity depends monotonically on the ratio ofligand concentrations. We then extend our analysis to more general schemes of com-binatorial control involving either additional binding sites for the two ligands or anadditional third ligand and show how these additions can cause a switch in the logicbehavior of the molecule. Overall, our results demonstrate the wide variety of controlschemes that biological systems can implement using simple mechanisms.
Introduction
A hallmark of cellular signaling and regulation is combinatorial control. Disparate exam-ples ranging from metabolic enzymes to actin polymerization to transcriptional regulationinvolve multiple inputs that often give rise to a much richer response than what could beachieved through a single-input. For example, the bacterial enzyme phosphofructokinase inthe glycolysis pathway is allosterically regulated by both ADP and PEP. Whereas PEPserves as an allosteric inhibitor, ADP is both an allosteric activator and a competitive in-hibitor depending upon its concentration. This modulation by multiple allosteric ligandsgives rise to a complex control of the flux through the glycolytic pathway: increasing ADPconcentration first increases the activity of phosphofructokinase (via the allosteric modula-tion) but ultimately decreases it (from competitive inhibition). Another example is offeredby the polymerization of actin at the leading edge of motile cells. In particular, the presenceof two ligands, Cdc42 and PIP2, is required to activate the protein N-WASP by bindingto it in a way that permits it to then activate the Arp2/3 complex and stimulate actinpolymerization. In the context of transcriptional regulation, an elegant earlier work explored the condi-tions under which transcriptional regulatory networks could give rise to the familiar Booleanlogic operations, like those shown in Figure 1. There it was found that the combined ef-fect of two distinct transcription factors on the transcriptional activity of a given promoterdepend upon their respective binding strengths as well as the cooperative interactions be-tween each other and the RNA polymerase. Indeed, by tuning the binding strengths andcooperativity parameters, one could generate a panoply of different logic gates such as thefamiliar AND, OR, NAND (NOT-AND) and NOR (NOT-OR) gates, known from the worldof digital electronics. Here we explore the diversity of combinatorial responses that can be effected by a singleallosteric molecule by asking if such molecules can yield multi-input combinatorial controlin the same way that transcriptional networks have already been shown to. Specifically, we2 L ][L ] [L ] [L ][L ][L ] [L ] [L ][L ] [L ] p a c t i v e p active [L ][L ] p active [L ][L ] p active p a c t i v e [L ] [L ] [L ] p a c t i v e [L ]LowLow 0 010 HighHighAND (A) (B) (C) LowLow 0 111 HighHighOR LowLow 0 101 HighHighXOR
Figure 1. Logic gates as molecular responses.
The (A) AND, (B) OR, and (C) XOR gatesare represented through their corresponding logic tables as well as target activity profiles regulatedby two ligands. The behavior of each gate is measured solely by its activity in the absence and atsaturating concentrations of each ligand and not by the character of the active/inactive transition. build on earlier work that shows that an allosteric molecule described by the Monod-Wyman-Changeux (MWC) model can deliver input-output functions similar to the ideal logic gatesdescribed in Figure 1.
In the MWC model, an allosteric molecule exists in a thermody-namic equilibrium between active and inactive states, with the relative occupancy of eachstate being modulated by regulatory ligands. We use statistical mechanics to characterizethe input-output response of such a molecule in the limits where each of the two ligands iseither absent or at a saturating concentration and determine the necessary conditions to formthe various logic gates, with our original contribution on this point focusing on a systematicexploration of the MWC parameter space for each logic gate.We then analyze the MWC response modulated by two input ligands but outside oftraditional Boolean logic functions. In particular, we show how, by tuning the MWC pa-rameters, the response (probability of the allosteric protein being active) in any three of thefour concentration limits can be explicitly controlled, along with the ligand concentrationsat which transitions between these limit responses occur. Focusing next on the profile ofthe response near the transition concentrations, we demonstrate how an MWC molecule canexhibit ratiometric sensing which was observed experimentally in the bone morphogeneticprotein (BMP) signaling pathway as well as in galactose metabolic (GAL) gene inductionin yeast. Additionally, we extend our analysis of logic responses to cases beyond two-ligand controlwith a single binding site for each ligand. We first discuss the effect of the number of bindingsites on the logic response and demonstrate how altering that number, which can occurthrough evolution or synthetic design, is able to cause a switch in the logic-behavior of anMWC molecule, such as transitioning from AND into OR behavior. Next, we explore theincreased diversity of logic responses that can be achieved by three-ligand MWC moleculescompared with the two-ligand case and offer an interesting perspective on the role of the thirdligand as a regulator that can switch the logic-behavior formed by the other two ligands. We3 ctivestate weight inactivestate weight1[L ]K A,1 [L ] ( ) K A,1 [L ] ( ) K A,2 [L ]K A,2 e -βΔε AI e -βΔε AI [L ]K I,1 e -βΔε AI [L ] ( ) K I,1 [L ] ( ) K I,2 e -βΔε AI [L ]K I,2 e -βΔε AI [L ]1+ ( ) K I,1 [L ]1+ ( ) K I,2 [L ]1+ ( ) K A,1 [L ]1+ ( ) K A,2
Figure 2. States and weights for the allosteric protein.
The two different ligands (bluecircle (i = 1 ) and red triangle (i = 2 )) are present at concentrations [ L i ] and with a dissociationconstant K A , i in the active state and K I , i in the inactive state. The energetic difference between theinactive and active states is denoted by ∆ ε AI = ε I − ε A . Total weights of the active and inactivestates are shown below each column and are obtained by summing all the weights in that column. end by a discussion of our theoretical results in the context of a growing body of experimentalworks on natural and de novo designed molecular logic gates. In total, these results hint atsimple mechanisms that biological systems can utilize to refine their combinatorial control. Results
Logic Response of an Allosteric Protein Modulated by Two Ligands
Consider an MWC molecule, as shown in Figure 2, that fluctuates between active and inactivestates (with ∆ ε AI defined as the free energy difference between the inactive and active statesin the absence of ligand). We enumerate the entire set of allowed states of activity andligand occupancy, along with their corresponding statistical weights. The probability thatthis protein is active depends on the concentrations of two input molecules, [ L ] and [ L ] ,and is given byp active ([ L ] , [ L ]) = (cid:16) [ L ] K A , (cid:17) (cid:16) [ L ] K A , (cid:17)(cid:16) [ L ] K A , (cid:17) (cid:16) [ L ] K A , (cid:17) + e − β ∆ ε AI (cid:16) [ L ] K I , (cid:17) (cid:16) [ L ] K I , (cid:17) , (1)where K A , i and K I , i are the dissociation constants between the i th ligand and the active orinactive protein, respectively. We begin with the two-input case such that i = 1 or .To determine whether this allosteric protein can serve as a molecular logic gate, we first4valuate the probability that it is active when each ligand is either absent ( [ L i ] → ) or ata saturating concentration ( [ L i ] → ∞ ). Figure 3A evaluates these limits for eq 1, where wehave introduced the parameters γ = K A , K I , and γ = K A , K I , to simplify the results.The probabilities in Figure 3A can be compared to the target functions in Figure 1 todetermine the conditions on each parameter that would be required to form a given logicgate. For example, the AND, OR, and XOR gates require that in the absence of either ligand( [ L ] = [ L ] = 0 ), there should be as little activity as possible, thereby requiring that theactive state has a higher (more unfavored) free energy than the inactive state (e − β ∆ ε AI (cid:29) ).We note that in the context of transcriptional regulation, this limit of activity in the absenceof ligands is called the leakiness, and it is one of the distinguishing features of the MWCmodel in comparison with other allosteric models such as the Koshland-Némethy-Filmer(KNF) model that exhibits no leakiness.For the AND and OR gates, the condition that p active ≈ when both ligands are satu-rating ( [ L ] , [ L ] → ∞ ) requires that γ γ e − β ∆ ε AI (cid:28) . The two limits where one ligand isabsent while the other ligand is saturating lead to the conditions shown in Figure 3B for theAND and OR gates, with representative response profiles shown in Figure 3C using param-eter values from the single-ligand allosteric nicotinic acetylcholine receptor. We relegatethe derivations to Appendix A, where we also demonstrate that the XOR gate cannot berealized with the form of p active in eq 1 unless explicit cooperativity is added to the MWCmodel. In addition, we show that the NAND, NOR, and XNOR gates can be formed ifand only if their complementary AND, OR, and XOR gates can be formed, respectively, byreplacing ∆ ε AI → − ∆ ε AI and γ i → γ i . Finally, Figure 3C demonstrates that the same dis-sociation constants K A , i and K I , i can give rise to either AND or OR behavior by modulating ∆ ε AI , with the transition between these two logic gates occurring at e − β ∆ ε AI ≈ γ ≈ γ (thiscorresponds to ∆ ε AI ≈ − k B T for the values of K A , i and K I , i in Figure 3).To explore the gating behavior changes across parameter space, we define a quality metricfor how closely p active matches its target value at different concentration limits for a givenidealized logic gate,Q ( γ , γ , ∆ ε AI ) = (cid:89) λ = 0 , ∞ (cid:89) λ = 0 , ∞ (1 − (cid:12)(cid:12) p ideal λ ,λ − p λ ,λ (cid:12)(cid:12) ) , (2)where p λ ,λ = p active ([ L ] → λ , [ L ] → λ ) . A value of 1 (high quality gate) implies aperfect match between the target function and the behavior of the allosteric molecule whilea value near 0 (low quality gate) suggests that the response behavior deviates from the targetfunction in at least one limit.From eq 2, the quality for the AND gate becomesQ AND = (1 − p , )(1 − p ∞ , )(1 − p , ∞ ) p ∞ , ∞ , (3)while for the OR gate it takes on the formQ OR = (1 − p , ) p ∞ , p , ∞ p ∞ , ∞ . (4)Figure 3D shows the regions in parameter space where the protein exhibits these gating be-5 A) (B)(C)
AND OR (D)
AND OR
ANDOR γ , γ << 1<< e -βΔε AI << 1γ γ γ , γ << 11 << e -βΔε AI << 1 ,γ
11 + e -βΔε AI
11 + γ e -βΔε AI
11 + γ e -βΔε AI
11 + γ γ e -βΔε AI p a c t i v e p a c t i v e γ γ Δε AI (k B T)Δε AI (k B T) 1.000.20.40.60.8quality [ L ] → [ L ] → ∞ [L ]→0 [L ]→∞ gate conditions [L ]K A,2 [L ]K A,1 [L ]K A,2 [L ]K A,1
Figure 3. Logic gate realization of an allosteric protein with two ligands. (A)Probability that the protein is active (p active ) in different limits (rows and columns of the matrix)of ligand concentrations, where γ i = K A , i K I , i . (B) Conditions on the parameters that lead to an ANDor OR response. (C) Realizations of the AND and OR logic gates. Parameters used wereK A , = K A , = 2 . × − M, K I , = K I , = 1 . × − M, and ∆ ε AI = − . k B T for the ANDgate or ∆ ε AI = − . k B T for the OR gate. (D) Quality of AND (eq 3) and OR (eq 4) gates acrossparameter space. The brown dots indicate the high quality gates in Panel C. haviors (the high quality gates from Figure 3C are denoted by brown dots). More specifically,for a fixed ∆ ε AI , the AND behavior is achieved in a finite triangular region in the γ - γ planewhich grows larger as ∆ ε AI decreases. The OR gate, on the other hand, is achieved in aninfinite region defined by γ , γ (cid:46) e β ∆ ε AI . In either case, a high quality gate can be obtainedonly when the base activity is very low ( ∆ ε AI (cid:46) ) and when both ligands are strong activa-tors ( γ , γ (cid:28) ), in agreement with the derived conditions (Figure 3B). Lastly, we note thatthe quality metrics for AND/OR and their complementary NAND/NOR gates obey a simple6elation, namely, Q AND/OR ( γ , γ , ∆ ε AI ) = Q NAND/NOR (cid:16) γ , γ , − ∆ ε AI (cid:17) , which follows fromthe functional form of eq 2 and the symmetry between the two gates (see Appendix A). General Two-Ligand MWC Response
We next relax the constraint that p active must either approach 0 or 1 in the limits of noligand or saturating ligand and consider the general behavior that can be achieved by anMWC molecule in the four limits shown in Figure 3A. Manipulating the three parameters( γ , γ and ∆ ε AI ) enables us to fix three of the four limits of p active , and these three choicesdetermine the remaining limit. For example, the parameters in Figure 4A were chosen sothat p , = 0 . ( ∆ ε AI = 0 ), p , ∞ ≈ . ( γ = 0 . ), and p ∞ , ≈ . ( γ = 20 ), which fixedp ∞ , ∞ ≈ . for the final limit. (B)(A) p a c t i v e [L ]K A,2 [L ]K A,1 p a c t i v e [L ]K A,2 [L ]K A,1 ratiometricresponseregion
Figure 4. General MWC response with two ligands. (A) Three of the four limits of ligandconcentrations ( [ L ] , [ L ] → or ∞ ) can be fixed by the parameters ∆ ε AI , γ , and γ .Additionally, the midpoint of the [ L i ] response when [ L j ] → (solid purple curve) or [ L j ] → ∞ (dashed purple curve) can be adjusted. (B) Within the region determined by the four midpoints,the MWC response becomes ratiometric where the concentration ratio of the two ligandsdetermines the activity of the molecule. This is illustrated by the diagonal contour lines ofconstant p active in the ratiometric response region. In addition to the limits of p active , the locations of the transitions between these limitscan be controlled by changing K A , i and K I , i while keeping γ i = K A , i K I , i constant. In Appendix Bwe generalize previous results for the transition of a single-ligand MWC receptor to thepresent case of two ligands. Interestingly, we find that the midpoint [ L ∗ ] [ L ] → of the responsein the absence of [ L ] (solid curve in Figure 4A) is different from the midpoint [ L ∗ ] [ L ] →∞ of the response at saturating [ L ] (dashed curve in Figure 4A), with analogous statementsholding for the second ligand. More precisely, the two transition points occur at [ L ∗ i ] [ L j ] → = K A , i e − β ∆ ε AI γ i e − β ∆ ε AI , (5) [ L ∗ i ] [ L j ] →∞ = K A , i γ j e − β ∆ ε AI γ γ e − β ∆ ε AI . (6)7otably, the ratio [ L ∗ i ] [ L j ] →∞ [ L ∗ i ] [ L j ] → = (1 + γ e − β ∆ ε AI )(1 + γ e − β ∆ ε AI )(1 + e − β ∆ ε AI )(1 + γ γ e − β ∆ ε AI ) (7)is invariant to ligand swapping (i ↔ j); hence, the transition zones, defined as the concen-tration intervals between solid and dotted curves, have identical sizes for the two ligands, ascan be seen in Figure 4.The MWC response has its steepest slope when the ligand concentration is within therange set by [ L ∗ i ] [ L j ] → and [ L ∗ i ] [ L j ] →∞ , and interesting response behaviors can arise when bothligand concentrations fall into this regime. For example, Antebi et al. recently showed thatthe BMP pathway exhibits ratiometric response where pathway activity depends mono-tonically on the ratio of the ligand concentrations. Similar response functions have alsobeen observed in the GAL pathway in yeast, where gene induction is sensitive to the ratioof galactose and glucose. Such behavior can be achieved within the highly sensitive re-gion of the MWC model using one repressor ligand (L ) and one activator ligand (L ), asshown in Figure 4B. Parameters chosen for demonstration are ∆ ε AI = 0 , K A , = K A , and K I , K A , = K A , K I , = 10 − . In this regime, the probability of the protein being active gets reducedto p active ([ L ] , [ L ]) ≈ [ L ] K A , [ L ] K A , + [ L ] K I , , (8)which clearly depends monotonically on the [ L ] / [ L ] ratio (see Appendix B for details). Wenote that the region over which the ratiometric behavior is observed can be made arbitrarilylarge by decreasing the ratios K I , K A , and K A , K I , . Modulation by Multiple Ligands
A much richer repertoire of signaling responses is available to an MWC protein if we gobeyond two ligand inputs with a single binding site for each, as exhibited by phosphofruc-tokinase, for example. Though earlier we mentioned phosphofructokinase in the context oftwo of its input ligands, in fact, this enzyme has even more inputs than that and thus pro-vides a rich example of multi-ligand combinatorial control. To start exploring the diversityof these responses, we generalize eq 1 to consider cases with N input ligands, where the i th ligand has n i binding sites, concentration [ L i ] , and dissociation constants K A , i and K I , i withthe molecule’s active and inactive states, respectively. In general, it is impractical to writethe states and weights as we have done in Figure 2, since the total number of possible states,given by (cid:80) Ni =1 n i , grows exponentially with the number of binding sites. However, by anal-ogy with the earlier simple case, the general formula for the probability that the protein isactive can be written asp active ([ L ] , [ L ] , ..., [ L N ]) = (cid:81) Ni =1 (cid:16) [ L i ] K A , i (cid:17) n i (cid:81) Ni =1 (cid:16) [ L i ] K A , i (cid:17) n i + e − β ∆ ε AI (cid:81) Ni =1 (cid:16) [ L i ] K I , i (cid:17) n i . (9)8e first consider an MWC molecule with N = 2 input ligands as in the previous sectionbut with n i ligand binding sites for ligand i. As derived in Appendix C, the criteria forthe AND and OR gates are identical to those for a protein with n i = 1 binding site perligand, except that we make the γ i → γ n i i substitution in the conditions shown in Figure 3B.The protein thus exhibits OR behavior if e − β ∆ ε AI (cid:28) min (cid:16) γ n , γ n (cid:17) or AND behavior ife − β ∆ ε AI (cid:29) max (cid:16) γ n , γ n (cid:17) .Over evolutionary time or through synthetic approaches, the number of binding sitesdisplayed by a single molecule can be tuned, enabling such systems to test a variety ofresponses with a limited repertoire of regulatory molecules. Since γ , γ (cid:28) , increasing thenumber of binding sites while keeping all other parameters the same can shift a responsefrom AND → OR as shown in Figure 5. The opposite logic switching (OR → AND) is similarlypossible by decreasing the number of binding sites, and analogous results can be derived forthe complementary NAND and NOR gates (see Appendix C). In the limit where the numberof binding sites becomes large (n , n (cid:29) ), an allosteric molecule’s behavior will necessarilycollapse into OR logic provided γ , γ < , since the presence of either ligand occupyingthe numerous binding sites has sufficient free energy to overcome the active-inactive freeenergy difference ∆ ε AI . In addition, having a large number of binding sites makes the p active response sharper (Figure 5B), as has been seen in the context of chromatin remodeling where ∼
150 bp of DNA “buried” within a nucleosome can be made available for transcription bythe binding of multiple transcription factors. (B) (A) conditionsγ , γ << 1 1 ,γ << e -βΔε AI <<1 ,γ AND → OR, increasing n , n p a c t i v e p a c t i v e [L ]K A,2 [L ]K A,1 [L ]K A,2 [L ]K A,1
Figure 5. Increased number of binding sites can switch the logic of an MWC proteinfrom AND into OR. (A) Parameter conditions required for AND → OR switching upon anincrease in the number of binding sites. (B) Representative activity plots showing the AND → ORswitching. Parameters used were K A , i = 2 . × − M, K I , i = 2 . × − M and ∆ ε AI = − k B T. Next, we examine an alternative possibility of generalizing the MWC response, namely,considering a molecule with N = 3 distinct ligands, each having a single binding site (n i = 1 ).The logic response is now described by a × × cube corresponding to the activity at lowand saturating concentrations of each of the three ligands (an example realization is shownin Figure 6A). Since each of the 8 cube elements can be either OFF or ON (red and greencircles, respectively), the total number of possible responses becomes = 256 . This number,however, includes functionally redundant responses, as well as ones that are not admissiblein the MWC framework. We therefore eliminate these cases in order to accurately quantifythe functional diversity of 3-input MWC proteins.9 NDORANDN i YES i NONE A N D O R Y E S i Y E S j N O N E N A N D N O R O R N i N O T i O R N j N O T j A LL A N D N i A N D N j L
122 233 23 ANDYES
132 3 L L i L j ( A ) ( B )( C ) L L L L ( D ) f un ct i ona ll yun i que MW C- c o m pa t i b l e MW C- i n c o m pa t i b l e f un ct i ona ll y r edundan t
80 104176
Figure 6. Third ligand expands the combinatorial diversity of logic responses andenables logic switching. (A) Cubic diagram of a representative molecular logic response. Thelabel “0” stands for the limit when all ligands are at low concentrations. Each digit in the labels ofother limits indicates the high concentration of the corresponding ligand (for example, in the “12”limit the ligands 1 and 2 are at high concentrations). Red and green colors indicate the OFF andON states of the molecule, respectively. (B) Diagram representing the numbers of 3-ligand logicgates categorized by their MWC compatibility and functional uniqueness. The area of each cell isproportional to the number of gates in the corresponding category. (C) Demonstration of differentlogic transitions induced by a third ligand (thick arrows) on the example of the 3-input gate inPanel A. (D) Table of all possible logic transitions (row → column, green cells) inducible by athird ligand in the MWC framework. Schematics of the 14 MWC-compatible 2-ligand gatescorresponding to each column entry are displayed on top (i and j represent different ligands).Results for the transitions between logical complements (NOT row → NOT column) are identicalto the results for row → column transitions and are not shown. Trivial transitions betweenidentical gates where the third ligand has no effect are marked with hatching lines. We consider two responses to be functionally identical if one can be obtained from anotherby relabeling the ligands, e.g. (1 , , → (3 , , . Eliminating all redundant responses leaves80 unique cases out of the 256 possibilities (see Appendix D). In addition, since the molecule’sactivity in the eight ligand concentration limits is determined by only four MWC parameters,10amely, { ∆ ε AI , γ , γ , γ } , we expect the space of possible 3-input gates to be constrained(analogous to XOR/XNOR gates being inaccessible to 2-input MWC proteins). Imposingthe constraints leaves 34 functionally unique logic responses that are compatible with theMWC framework (see Figure 6B for the summary statistics and Appendix D for the detaileddiscussion of how the constraints were imposed).In addition to expanding the scope of combinatorial control relative to the two-inputcase, we can think of the role of the third ligand as a regulator whose presence switchesthe logic performed by the other two ligands. We illustrate this role in Figure 6C by firstfocusing on the leftmost cubic diagram. The gating behavior on the left face of the cube (inthe absence of L ) exhibits NONE logic while the behavior on the right face of the cube (inthe presence of saturating L ) is the ORN logic (see the schematics at the top of Figure 6Dfor the definition of all possible gates). In this way, adding L switches the logic of theremaining two ligands from NONE → ORN . In a similar vein, adding L changes the logicfrom ANDN → YES , while adding L causes a YES → AND switch.We repeat the same procedure for all functionally unique 3-ligand MWC gates (see Ap-pendix D) and obtain a table of all possible logic switches that can be induced by a thirdligand (green cells in Figure 6D that indicate row → column logic switches). As we can see,a large set of logic switches are feasible, the majority of which (the left half of the table) donot involve a change in the base activity (i.e., activity in the absence of the two ligands).Comparatively fewer transitions that involve flipping of the base activity from OFF to ONare possible (the right half of the table).As a demonstration of the regulatory function of the third ligand, we show two exam-ples of logic switching induced by increasing [L ], namely, AND → OR (Figure 7A,B) andAND → YES (Figure 7C,D), along with the parameter conditions that need to be satisfiedto enable such transitions (see Appendix D for derivations). An interesting perspective is toview the L ligand as a modulator of the free energy difference ∆ ε AI . For example, when [ L ] = 0 , the protein behaves identically to the N = 2 case given by eq 1; at a saturatingconcentration of L , however, the protein behaves as if it had N = 2 ligands with a modifiedfree energy difference ∆ ε (cid:48) AI given by ∆ ε (cid:48) AI = ∆ ε AI − k B T log γ . (10)From this perspective, the third ligand increases the effective free energy difference in theexamples shown in Figure 7, since in both cases the γ (cid:28) condition is satisfied. Forthe AND → OR transition, the increase in ∆ ε AI is sufficient to let either of the two ligandsactivate the molecule (hence, the OR gate). In the AND → YES transition, the change in ∆ ε AI utilizes the asymmetry between the binding strengths of the two ligands ( γ (cid:28) γ ) toeffectively “silence” the activity of the ligand L . We note in passing that such behavior forthe N = 3 allosteric molecule is reminiscent of a transistor which can switch an input signalin electronics. 11 onditions (B)(A) AND → OR, increasing [L ] p a c t i v e p a c t i v e γ i γ j γ k << << conditions [L ] = 0 [L ] = 10 K A,3 [L ] = 0 [L ] = 10 K A,3 conditions (D)(C)
AND → YES , increasing [L ] p a c t i v e p a c t i v e conditions γ , γγ << << e - β Δ ε A I γ γ γ γ γ , γ γ , << << e - β Δ ε A I γ j γ k γ k γ i γ k , << << [L ]K A,2 [L ]K A,1 [L ]K A,2 [L ]K A,1 [L ]K A,2 [L ]K A,1 [L ]K A,2 [L ]K A,1 ( ≤ i , j , k ≤ ) Figure 7. Example logic switches induced by the third ligand.
Parameter conditions andrepresentative activity plots of an allosteric molecule exhibiting AND logic in the absence of thethird ligand, while exhibiting OR logic (A,B) or YES logic (C,D) when L is present at asaturating concentration. Parameters used were K A , i = 2 . × − M and K I , i = 2 . × − M inPanel B, K A , i = 2 . × − M, K I , = 2 . × − M and K I , = 2 . × − M in panel D, alongwith ∆ ε AI = − k B T in both panels.
Discussion and Conclusions
Combinatorial control is a ubiquitous strategy employed by cells. Networks of cellular sys-tems of different kinds, such as transcriptional, signaling, or metabolic, integrate infor-mation from multiple inputs in order to produce a single output. The statistical mechanicalMWC model we employ allows us to systematically explore the combinatorial diversity ofoutput responses available to such networks and determine the conditions that the MWCparameters need to satisfy to realize a particular response.In this paper, we built on earlier work to show that the response of an allosteric MWCmolecule can mimic Boolean logic. Specifically, we demonstrated that a protein that bindsto two ligands can exhibit an AND, OR, NAND, or NOR response (also shown by others ),where the former two cases require the protein to be inherently inactive and that both lig-ands preferentially bind to the active conformation, whereas the latter two cases require theconverse conditions. We derived the MWC parameter ranges within which an allosteric pro-tein would exhibit an AND or OR response (Figure 3B), and showed that the correspondingparameter ranges for NAND or NOR responses could be achieved by simply substituting γ i → γ i and ∆ ε AI → − ∆ ε AI in the parameter condition equations (Appendix A.3). Sincethe NAND and NOR gates are known in digital electronics as universal logic gates, all otherlogic functions can be reproduced by hierarchically layering these gates. In the context ofthis work, such layering could be implemented if the MWC protein is an enzyme that only12atalyzes in the active state so that its output (the amount of product) could serve as aninput for the next enzyme, thereby producing more complex logic functions via allostery,though at the cost of noise amplification and response delays.As in earlier work, we showed that the XOR and XNOR responses cannot be achievedwithin the original MWC framework (eq 1) but are possible when cooperativity betweenthe two ligands is introduced (Appendix A.4). Biological XOR and XNOR behaviors areuncommon in non-transcriptional systems and have also been challenging for synthetic designand optimization. One of the few examples of such systems is a synthetic metallochromicchromophore whose transmittance output level is modulated by Ca and H + ions in aXOR-like manner. In addition to traditional Boolean logic, we recognized further manifestations of com-binatorial control by two-ligand MWC proteins. In particular, we showed that the proteinactivity in three of the four ligand concentration limits can be set independently by tuning theMWC parameters γ , γ , and ∆ ε AI , and that the ligand concentrations at which transitionsbetween limit responses take place can be separately controlled by proportionally changingK A , i and K I , i , while keeping γ i = K A , i K I , i constant (eqs 5 and 6). We also showed that when theranges of ligand concentrations are close to those transition values, then ratiometric sensingobserved in the BMP and GAL pathways, can be recapitulated through the MWC model(Figure 4B), with larger regions of sensitivity achievable by an appropriate tuning of theparameters. We note that parameter “tuning” can be realized either through evolutionaryprocesses over long time scales or synthetically, using mutagenesis or other approaches. Apart from altering the thermodynamic parameters such as the ligand binding affinity orthe free energy of active and inactive protein conformations, the number of ligand bindingsites of an allosteric molecule can also be changed. This can occur evolutionarily throughrecombination events, synthetically by engineering combinations of protein domains, orthrough binding of competitive effectors that reduce the effective number of ligand bindingsites. We found that these alterations in the number of ligand binding sites are capable ofswitching the logic behavior between AND ↔ OR or NAND ↔ NOR gates (Figure 5B). Sincethe MWC model has even been applied in unusual situations such as the packing of DNA intonucleosomes, these results on combinatorial control can also be relevant for eukaryotictranscription. The opening of the nucleosome is itself often subject to combinatorial controlbecause there can be multiple transcription factor binding sites within a given nucleosome,the number of which can also be tuned using synthetic approaches.
Lastly, we generalized the analysis of logic responses for a molecule whose activity ismodulated by three ligands, and identified 34 functionally unique and MWC-compatiblegates out of 256 total possibilities. We offered a perspective on the function of any of thethree ligands as a “regulator” that can cause a switch in the type of logic performed by theother two ligands and derived the full list of such switches (Figure 6D). Within the MWCmodel, the role of this regulatory ligand can be viewed as effectively changing the free energydifference ∆ ε AI between the protein’s active and inactive states (Appendix D.2), which, inturn, is akin to the role of methylation or phosphorylation in adaptation, but withoutthe covalent linkage. Our in-depth analysis of the logic repertoire available to 3-input MWCmolecules can serve as a theoretical framework for designing new allosteric proteins and alsofor understanding the measured responses of existing systems. Examples of such systems13hat both act as 3-input AND gates include the GIRK channel, the state of which (open orclosed) is regulated by the G protein G β γ , the lipid PIP and Na + ions, or the engineeredN-WASP signaling protein which is activated by SH3, Cdc42 and PDZ ligands. The exquisite control that arises from the web of interactions underlying biological sys-tems is difficult to understand and replicate. A first step to overcoming this hurdle is tocarefully quantify the types of behaviors that can arise from multi-component systems. Asour ability to harness and potentially design de novo allosteric systems grows, we canaugment our current level of combinatorial control in biological contexts, such as transcrip-tional regulation, to create even richer dynamics.
Acknowledgements
It is a great pleasure to acknowledge the contributions of Bill Eaton to our understand-ing of allostery. We thank Chandana Gopalakrishnappa and Parijat Sil for their input onthis work, and Michael Elowitz for his insights and valuable feedback on the manuscript.This research was supported by La Fondation Pierre-Gilles de Gennes, the Rosen Center atCaltech, the Department of Defense through the National Defense Science & EngineeringGraduate Fellowship (NDSEG) Program (LF), and the National Institutes of Health DP1OD000217 (Director’s Pioneer Award), R01 GM085286, and 1R35 GM118043-01 (MIRA).We are grateful to the Burroughs-Wellcome Fund for its support of the Physical Biology ofthe Cell Course at the Marine Biological Laboratory, where part of this work was completed.
References (1) Blangy, D.; Buc, H.; Monod, J. Kinetics of the Allosteric Interactions of Phosphofruc-tokinase from
Escherichia Coli . Journal of Molecular Biology , , 13–35.(2) Dueber, J. E.; Yeh, B. J.; Chak, K.; Lim, W. A. Reprogramming Control of an AllostericSignaling Switch Through Modular Recombination. Science , , 1904–1908.(3) Buchler, N. E.; Gerland, U.; Hwa, T. On Schemes of Combinatorial Transcription Logic. Proceedings of the National Academy of Sciences , , 5136–41.(4) Graham, I.; Duke, T. The Logical Repertoire of Ligand-Binding Proteins. PhysicalBiology , , 159–165.(5) de Ronde, W.; ten Wolde, P. R.; Mugler, A. Protein Logic: A Statistical MechanicalStudy of Signal Integration at the Single-Molecule Level. Biophysical Journal , , 1097–1107.(6) Agliari, E.; Altavilla, M.; Barra, A.; Schiavo, L. D.; Katz, E. Notes on Stochastic(Bio)-Logic Gates: Computing With Allosteric Cooperativity. Scientific Reports , , 9415.(7) Martins, B. M.; Swain, P. S. Trade-offs and Constraints in Allosteric Sensing. PLoSComputational Biology , , e1002261.148) Antebi, Y. E.; Linton, J. M.; Klumpe, H.; Bintu, B.; Gong, M.; Su, C.; McCardell, R.;Elowitz, M. B. Combinatorial Signal Perception in the BMP Pathway. Cell , ,1184–1196.(9) Escalante-Chong, R.; Savir, Y.; Carroll, S. M.; Ingraham, J. B.; Wang, J.; Marx, C. J.;Springer, M. Galactose Metabolic Genes in Yeast Respond to a Ratio of Galactose andGlucose. Proceedings of the National Academy of Sciences , , 1636–1641.(10) Razo-Mejia, M.; Barnes, S. L.; Belliveau, N. M.; Chure, G.; Einav, T.; Lewis, M.;Phillips, R. Tuning Transcriptional Regulation through Signaling: A Predictive Theoryof Allosteric Induction. Cell Systems , , 456–469.(11) Auerbach, A. Thinking in Cycles: MWC Is a Good Model for Acetylcholine Receptor-Channels. The Journal of Physiology , , 93–98.(12) Marzen, S.; Garcia, H. G.; Phillips, R. Statistical Mechanics of Monod-Wyman-Changeux (MWC) Models. Journal of Molecular Biology , , 1433–1460.(13) Mirny, L. A. Nucleosome-Mediated Cooperativity Between Transcription Factors. Pro-ceedings of the National Academy of Sciences , , 22534–22539.(14) Scholes, C.; DePace, A. H.; Sánchez, Á. Combinatorial Gene Regulation through KineticControl of the Transcription Cycle. Cell Systems , , 97–108.(15) Kinkhabwala, A.; Guet, C. C. Uncovering Cis Regulatory Codes Using Synthetic Pro-moter Shuffling. PLoS ONE , , e2030.(16) Dueber, J. E.; Yeh, B. J.; Bhattacharyya, R. P.; Lim, W. A. Rewiring Cell Signaling:The Logic and Plasticity of Eukaryotic Protein Circuitry. Current Opinion in StructuralBiology , , 690–699.(17) Privman, V.; Zhou, J.; Halámek, J.; Katz, E. Realization and Properties of Biochemical-Computing Biocatalytic XOR Gate Based on Signal Change. The Journal of PhysicalChemistry B , , 13601–13608.(18) de Silva, A. P.; McClenaghan, N. D. Simultaneously Multiply-Configurable or Super-posed Molecular Logic Systems Composed of ICT (Internal Charge Transfer) Chro-mophores and Fluorophores Integrated with One- or Two-Ion Receptors. Chemistry-AEuropean Journal , , 4935–4945.(19) De Silva, A. P.; Uchiyama, S. Molecular Logic and Computing. Nature Nanotechnology , , 399.(20) Bloom, J. D.; Meyer, M. M.; Meinhold, P.; Otey, C. R.; MacMillan, D.; Arnold, F. H.Evolving Strategies for Enzyme Engineering. Current Opinion in Structural Biology , , 447–452.(21) Guntas, G.; Ostermeier, M. Creation of an Allosteric Enzyme by Domain Insertion. Journal of Molecular Biology , , 263–273.1522) Narula, J.; Igoshin, O. A. Thermodynamic Models of Combinatorial Gene Regulationby Distant Enhancers. IET Systems Biology , , 393–408.(23) Löhr, U.; Chung, H.-R.; Beller, M.; Jäckle, H. Antagonistic Action of Bicoid and theRepressor Capicua Determines the Spatial Limits of Drosophila
Head Gene ExpressionDomains.
Proceedings of the National Academy of Sciences , , 21695–21700.(24) Fakhouri, W. D.; Ay, A.; Sayal, R.; Dresch, J.; Dayringer, E.; Arnosti, D. N. Deci-phering a Transcriptional Regulatory Code: Modeling Short-Range Repression in the Drosophila
Embryo.
Molecular systems biology , , 341.(25) Chen, H.; Xu, Z.; Mei, C.; Yu, D.; Small, S. A System of Repressor Gradients SpatiallyOrganizes the Boundaries of Bicoid-Dependent Target Genes. Cell , , 618–629.(26) Crocker, J.; Tsai, A.; Stern, D. L. A Fully Synthetic Transcriptional Platform for aMulticellular Eukaryote. Cell Reports , , 287–296.(27) Hansen, C. H.; Endres, R. G.; Wingreen, N. S. Chemotaxis in Escherichia Coli : AMolecular Model for Robust Precise Adaptation.
PLoS Computational Biology , , e1.(28) Lan, G.; Sartori, P.; Neumann, S.; Sourjik, V.; Tu, Y. The Energy-Speed-AccuracyTrade-Off in Sensory Adaptation. Nature Physics , , 422.(29) Wang, W.; Touhara, K. K.; Weir, K.; Bean, B. P.; MacKinnon, R. Cooperative Reg-ulation by G Proteins and Na(+) of Neuronal GIRK2 K(+) Channels. eLife , ,e157519.(30) Dueber, J. E.; Mirsky, E. A.; Lim, W. A. Engineering Synthetic Signaling Proteins withUltrasensitive Input/Output Control. Nature Biotechnology , , 660–662.(31) Raman, A. S.; White, K. I.; Ranganathan, R. Origins of Allostery and Evolvability inProteins: A Case Study. Cell , , 468–480.(32) Huang, P. S.; Boyken, S. E.; Baker, D. The Coming of Age of De Novo
Protein Design.
Nature , , 320.(33) Guntas, G.; Mansell, T. J.; Kim, J. R.; Ostermeier, M. Directed Evolution of ProteinSwitches and Their Application to the Creation of Ligand-Binding Proteins. Proceedingsof the National Academy of Sciences , , 11224–11229.(34) Wei, H.; Hu, B.; Tang, S.; Zhao, G.; Guan, Y. Repressor Logic Modules Assembledby Rolling Circle Amplification Platform to Construct a Set of Logic Gates. ScientificReports , , 37477.(35) Macía, J.; Posas, F.; Solé, R. Distributed Computation: The New Wave of SyntheticBiology Devices. Trends in Biotechnology , , 342–349.16 upporting Information Contents A Derivation of Conditions for Achieving Different Logic Responses S2
A.1 AND Gate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S3A.2 OR Gate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S3A.3 NAND and NOR Gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S3A.4 XOR and XNOR Gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S4
B The General Two-Ligand Response: Transitioning Between OFF and ONStates S6C Logic Switching by Tuning the Number of Ligand Binding Sites S9D Combinatorial Control with Three Regulatory Ligands S10
D.1 Functionally Unique MWC Gates . . . . . . . . . . . . . . . . . . . . . . . . S10D.2 Logic Switching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S12S1
Derivation of Conditions for Achieving Different LogicResponses
In this section we derive the conditions necessary for an MWC molecule modulated by twoligands (with one binding site for each ligand) to exhibit the behavior of various logic gatesshown in Figure 1. In addition to the three logic gates shown in Figure 1, we will also discussthe three complimentary gates NAND, NOR, and XNOR depicted in Figure S1. [L ][L ] [L ] [L ][L ][L ] [L ] [L ][L ] [L ] p a c t i v e p active [L ][L ] p active [L ][L ] p active p a c t i v e [L ] [L ] [L ] p a c t i v e [L ]LowLow 1 101 HighHighNAND (A) (B) (C) LowLow 1 000 HighHighNOR LowLow 1 010 HighHighXNOR
Figure S1. Additional logic gates as molecular responses.
The (A) NAND, (B) NOR, and(C) XNOR gates are the compliments of the AND, OR, and XOR gates, respectively, shown inFigure 1.
To simplify our notation, we define the value of p active from eq 1 in the following limits,p , = p active ([ L ] → , [ L ] →
0) = 11 + e − β ∆ ε AI , (S1)p ∞ , = p active ([ L ] → ∞ , [ L ] →
0) = 11 + γ e − β ∆ ε AI , (S2)p , ∞ = p active ([ L ] → , [ L ] → ∞ ) = 11 + γ e − β ∆ ε AI , (S3)p ∞ , ∞ = p active ([ L ] → ∞ , [ L ] → ∞ ) = 11 + γ γ e − β ∆ ε AI , (S4)where γ i = K A , i K I , i is the ratio of the dissociation constants between the i th ligand and theprotein in the active and inactive states. From the ideal logic gate behaviors visualized inFigure 1 and Figure S1, we can then deduce the desired constraints that model parametersneed to meet for an effective realization of each gate.S2 .1 AND Gate Starting from the AND gate, we require p , ≈ , p , ∞ ≈ , p ∞ , ≈ and p ∞ , ∞ ≈ , whichyields the following conditions: e − β ∆ ε AI (cid:29) , (S5) γ e − β ∆ ε AI (cid:29) , (S6) γ e − β ∆ ε AI (cid:29) , (S7) γ γ e − β ∆ ε AI (cid:28) . (S8)Combining eqs S6-S8, we obtain the condition for an AND gate, namely, γ , γ (cid:28) e − β ∆ ε AI (cid:28) γ γ . (S9)Note, that the outer inequalities imply γ , γ (cid:28) , (S10)meaning that both ligands bind more tightly to the protein in the active than the inactivestate. A.2 OR Gate
For p active to represent an OR gate across ligand concentration space, it must satisfy p , ≈ ,p , ∞ ≈ , p ∞ , ≈ and p ∞ , ∞ ≈ . This requires that the parameters obeye − β ∆ ε AI (cid:29) , (S11) γ e − β ∆ ε AI (cid:28) , (S12) γ e − β ∆ ε AI (cid:28) , (S13) γ γ e − β ∆ ε AI (cid:28) . (S14)Combining eqs S11-S13, we obtain a constraint on the free energy difference, (cid:28) e − β ∆ ε AI (cid:28) γ , γ . (S15)As with the AND gate, the outer inequalities imply that the ligands prefer binding to theprotein in the active state, γ , γ (cid:28) . (S16) A.3 NAND and NOR Gates
Because the NAND and NOR gates are the logical complements of AND and OR gates,respectively, the parameter constraints under which they are realized are the opposites ofS3hose for AND and OR gates. Hence, the conditions for a NAND gate are given by γ γ (cid:28) e − β ∆ ε AI (cid:28) γ , γ (S17)while the conditions for NOR gates are γ , γ (cid:28) e − β ∆ ε AI (cid:28) . (S18)We note that in both cases, the outer inequalities imply that both ligands bind more tightlyto the protein in the inactive state than in the active state, γ , γ (cid:29) .The symmetry between AND/OR and NAND/NOR gates also implies a simple relationbetween their quality metrics, namely, Q AND/OR ( γ , γ , ∆ ε AI ) = Q NAND/NOR (cid:16) γ , γ , − ∆ ε AI (cid:17) .Here we provide a proof for the AND gate and invite the reader to do the same for the ORgate. From eq 2, the quality metrics for the AND and NAND gates can be written asQ AND ( γ , γ , ω ) = (1 − p , )(1 − p ∞ , )(1 − p , ∞ ) p ∞ , ∞ = (cid:18) −
11 + ω (cid:19) (cid:18) −
11 + γ ω (cid:19) (cid:18) −
11 + γ ω (cid:19) (cid:18)
11 + γ γ ω (cid:19) = γ γ ω (1 + ω )(1 + γ ω )(1 + γ ω )(1 + γ γ ω ) , (S19)Q NAND ( γ , γ , ω ) = p , p ∞ , p , ∞ (1 − p ∞ , ∞ )= (cid:18)
11 + ω (cid:19) (cid:18)
11 + γ ω (cid:19) (cid:18)
11 + γ ω (cid:19) (cid:18) −
11 + γ γ ω (cid:19) = γ γ ω (1 + ω )(1 + γ ω )(1 + γ ω )(1 + γ γ ω ) , (S20)where we introduced ω = e − β ∆ ε AI . Substituting γ → γ − , γ → γ − , ω → ω − (equivalentto ∆ ε AI → − ∆ ε AI ) in eq S20, we obtainQ NAND ( γ − , γ − , ω − ) = γ − γ − ω − (1 + ω − )(1 + γ − ω − )(1 + γ − ω − )(1 + γ − γ − ω − ) × γ γ ω γ γ ω = γ γ ω (1 + ω )(1 + γ ω )(1 + γ ω )(1 + γ γ ω ) ≡ Q AND ( γ , γ , ω ) . (S21) A.4 XOR and XNOR Gates
Here, we show that the XOR gate (and by symmetry the XNOR gate) are not achievablewith the form of p active given in eq 1. An XOR gate satisfies p , ≈ , p , ∞ ≈ , p ∞ , ≈ and p ∞ , ∞ ≈ which necessitates the parameter conditionse − β ∆ ε AI (cid:29) , (S22)S4 e − β ∆ ε AI (cid:28) , (S23) γ e − β ∆ ε AI (cid:28) , (S24) γ γ e − β ∆ ε AI (cid:29) . (S25)However, these conditions cannot all be satisfied, as the left-hand side of eq S25 can bewritten in terms of the left-hand sides of eqs S22-S24, γ γ e − β ∆ ε AI = (cid:0) γ e − β ∆ ε AI (cid:1) (cid:0) γ e − β ∆ ε AI (cid:1) e − β ∆ ε AI (cid:28) , (S26)contradicting eq S25.The XOR gate could be realized if an explicit cooperativity energy ε A,coop is added whenboth ligands are bound in the active state and ε I,coop when both are bound in the inactivestate. These cooperative interactions modify eq 1 to the formp active ([ L ] , [ L ]) = 1 + [ L ] K A , + [ L ] K A , + [ L ] K A , [ L ] K A , e − βε A,coop [ L ] K A , + [ L ] K A , + [ L ] K A , [ L ] K A , e − βε A,coop + e − β ∆ ε AI (cid:16) [ L ] K I , + [ L ] K I , + [ L ] K I , [ L ] K I , e − βε I,coop (cid:17) . (S27)Figure S2 demonstrates that the same parameter values from Figure 3B together with the(unfavorable) cooperativity energy ε A,coop = 15 k B T and ε I,coop = 0 can create an XOR gate. p a c t i v e [L ]K A,2 [L ]K A,1
Figure S2. An XOR gate can be achieved by adding cooperativity.
The activity profiledefined in eq S27 for the parameter values from Figure 3B, along with the cooperativity energies ε A,coop = 15 k B T and ε I,coop = 0 , give rise to an XOR response. S5 The General Two-Ligand Response: Transitioning Be-tween OFF and ON States
In the preceding section, we have been solely concerned with the behavior of the MWCmolecule in the limits of ligand concentration ( [ L i ] = 0 and [ L i ] → ∞ ), and have ignoredthe details about the transition from ON to OFF (e.g., its shape and steepness) and alsothe possibility of p active (cid:54) = 0 or . In this section, we examine and derive in greater detailsome of the additional response behaviors that are possible for an MWC molecule regulatedwith N = 2 ligands when the locations of transitions between limit responses are taken intoaccount.To examine the transitions between p active levels, we derive expressions for the concen-trations at which transitions are at their midpoint. Since p active is a function of two differentligand concentrations, [L ] and [L ], we define two different midpoint concentrations of ligandL i : one in the absence of ligand L j , [ L ∗ i ] [ L j ] → , and another when L j is saturating, [ L ∗ i ] [ L j ] →∞ .In particular, [ L ∗ i ] [ L j ] → is defined such thatp active (cid:0) [ L ∗ i ] [ L j ] → , [ L j ] = 0 (cid:1) = p active ([ L i ] = 0 , [ L j ] = 0) + p active ([ L i ] → ∞ , [ L j ] = 0)2 , (S28)i.e., the concentration of ligand i where p active is equal to the mean of the two p active limitvalues being transitioned between. If we evaluate the left hand side of eq S28 with i = 1 andj = 2 using eq 1, and the right hand side using the limits from Figure 3(A), we obtain (cid:16) [ L ∗ ] [ L2 ] → K A , (cid:17)(cid:16) [ L ∗ ] [ L2 ] → K A , (cid:17) + e − β ∆ ε AI (cid:16) [ L ∗ ] [ L2 ] → K I , (cid:17) = 12 (cid:18)
11 + e − β ∆ ε AI + 11 + γ e − β ∆ ε AI (cid:19) . (S29)Introducing γ = K A , / K I , , we can solve for [ L ∗ ] [ L ] → to find [ L ∗ ] [ L ] → K A , = 1 + e − β ∆ ε AI γ e − β ∆ ε AI . (S30)Eq S30 can be rewritten for [ L ∗ ] [ L ] → by merely interchanging all ligand and parameterindices, i.e., 1 ↔ [ L ∗ i ] [ L j ] →∞ we can re-write S28 using eq 1 in the casethat [L j ] → ∞ with i = 1 and j = 2, resulting in (cid:16) [ L ∗ ] [ L2 ] →∞ K A , (cid:17)(cid:16) [ L ∗ ] [ L2 ] →∞ K A , (cid:17) + γ e − β ∆ ε AI (cid:16) [ L ∗ ] [ L2 ] →∞ K I , (cid:17) = 12 (cid:18)
11 + γ e − β ∆ ε AI + 11 + γ γ e − β ∆ ε AI (cid:19) . (S31)S6q S31 can be solved for [ L ∗ ] [ L ] →∞ to produce, [ L ∗ ] [ L ] →∞ K A , = 1 + γ e − β ∆ ε AI γ γ e − β ∆ ε AI . (S32)Again, the symmetric expression for [ L ∗ ] [ L ] →∞ is found by swapping ligand and parameterindices, 1 ↔ et al. , which was briefly discussed earlier. Specifically, this responsecan be approximated by choosing parameter values that satisfy two desired limits, p ∞ , ≈ ( γ e − β ∆ ε AI (cid:29) ) and p , ∞ ≈ ( γ e − β ∆ ε AI (cid:28) ), as well as produce a large transition regionsensitive to both ligands, i.e., the ratio in eq 7, [ L ∗ i ] [ Lj ] →∞ [ L ∗ i ] [ Lj ] → is far from 1. One way to satisfythese conditions is to set K I , (cid:29) K A , = K A , (cid:29) K I , and ∆ ε AI = 0 in eq 1. Notice that withthese parameter choices and provided the ligand concentrations satisfy [ L ] K A , , [ L ] K I , (cid:28) , [ L ] K I , , [ L ] K A , (cid:29) , (S33)the probability that the protein is active reduces top active ([ L ] , [ L ]) ≈ [ L ] K A , [ L ] K A , + [ L ] K I , . (S34)Hence, only the ratio of [ L ] and [ L ] matters, as shown in Figure 4B where eq S33 is satisfiedprovided that − (cid:46) [ L ] K A , (cid:46) (cid:46) [ L ] K A , (cid:46) .Additionally, we consider the remaining three types of input-output computations shownby Antebi et al. to exist in the BMP pathway which they called the additive, imbalance,and balance responses. The additive response (which responds more to larger input concen-trations) is an OR gate which we showed is possible in Figure 3B. The imbalance response(which responds maximally to extreme ratios of the two input ligands) is similar to an XORbehavior which, as discussed in Appendix A.4, is only achievable with an explicit coopera-tivity energy.The balance response is defined asp balanceactive = (cid:40) L ] ≈ [ L ]0 [ L ] (cid:54)≈ [ L ] (S35)so that the protein is only ON when both ligands are present in the same amount as shownin Figure S3A. Such behavior is not possible within the MWC model because starting fromany point [ L ] = [ L ] , p active in eq 1 must either monotonically increase or monotonicallydecrease with [ L ] (depending on γ ), whereas eq S35 requires that p active must decrease forS7oth [ L ] > [ L ] and [ L ] < [ L ] (with similar contradictory statements for [ L ] ). The closestbehavior achievable by the MWC model is to zoom into the transition region of an XNORgate as shown in Figure S3B. As we zoom out of the concentration ranges shown, the foursquare regions of the plot will continue to expand as squares and the behavior will no longerapproximate the ideal balance response. p a c t i v e [L ]K A,2 [L ]K A,1 p a c t i v e [L ]K A,2 [L ]K A,1 (B)(A)
Figure S3. Balance response behavior approximated by the MWC model. (A) Theideal balance response from the BMP pathway and (B) the closest behavior that an MWCmolecule can exhibit using the complementary parameters from Figure S2 (K A , i = 1 . × − M,K I , i = 2 . × − M, ∆ ε AI = 5 k B T, ε A,coop = − k B T and ε I,coop = 0 ). S8 Logic Switching by Tuning the Number of Ligand Bind-ing Sites
In this section, we show how an MWC molecule whose activity is given by eq 9 can switchbetween exhibiting AND ↔ OR or NAND ↔ NOR behaviors by tuning the number of bindingsites. To begin, we define the probability p active that the molecule is active in the case whenthe i th ligand has n i binding sites, namely,p , = p active ([ L ] → , [ L ] →
0) = 11 + e − β ∆ ε AI , (S36)p ∞ , = p active ([ L ] → ∞ , [ L ] →
0) = 11 + γ n e − β ∆ ε AI , (S37)p , ∞ = p active ([ L ] → , [ L ] → ∞ ) = 11 + γ n e − β ∆ ε AI , (S38)p ∞ , ∞ = p active ([ L ] → ∞ , [ L ] → ∞ ) = 11 + γ n γ n e − β ∆ ε AI . (S39)Note that the only effect of having an arbitrary number of ligand binding sites (as opposedto n i = 1 as in Appendix A) is that the ratio of dissociation constants always appears raisedto the number of binding sites, γ n i i . Hence, the parameter conditions derived for AND andOR behaviors for n i = 1 can be used in the case of general n i by substituting γ i → γ n i i .Now, suppose a molecule with N = 2 ligands and with n (cid:48) and n (cid:48) binding sites for ligands1 and 2 represents an AND gate, while this same molecule with n and n binding sites servesas an OR gate, as in Figure 5B with n (cid:48) = n (cid:48) = 1 and n = n = 4 . From Figure 3B, theconditions in the former case (AND gate) are γ n (cid:48) , γ n (cid:48) (cid:28) e − β ∆ ε AI (cid:28) γ n (cid:48) γ n (cid:48) , (S40)while the conditions in the latter case (OR gate) are (cid:28) e − β ∆ ε AI (cid:28) γ n , γ n . (S41)Combining these conditions, we find that the requirements for the AND ↔ OR switching aregiven by γ n (cid:48) , γ n (cid:48) (cid:28) e − β ∆ ε AI (cid:28) γ n , γ n , γ n (cid:48) γ n (cid:48) , (S42)where we have used the fact that the outer inequalities imply γ n (cid:48) , γ n (cid:48) (cid:28) (so that (cid:28) γ n (cid:48) , γ n (cid:48) ). In the limit n (cid:48) = n (cid:48) = 1 , eq S42 reduces to the condition shown in Figure 5A.Lastly, we note that since NAND is the complement of AND while NOR is the complementof OR, the class switching requirements in S42 become the requirements to change fromNAND ↔ NOR behavior when γ i → γ i and ∆ ε AI → − ∆ ε AI .S9 Combinatorial Control with Three Regulatory Ligands
In this section, we first present the methodology used to identify the functionally uniqueand MWC-compatible 3-ligand logic gates. We then use the full list of admissible gates tofind all possible logic switches that can be induced by increasing the concentration of a thirdligand. We finish the section by deriving the parameter conditions required for achieving thelogic switches AND → OR and AND → YES shown in Figure 7D. D.1 Functionally Unique MWC Gates
To identify the set of functionally unique MWC gates, we first iterate over the 256 possibleresponses and eliminate those redundant ones that can be obtained by shuffling the ligandlabels of already selected gates. The Python implementation of this procedure that leaves80 functionally unique gates can be found in the supplementary Jupyter Notebook 1.Having singled out the functionally unique responses, we proceed to identify those thatare admissible in the MWC framework. To that end, we first write the analytic forms for theprobability of the protein being active (p active ) at eight different ligand concentration limits(Figure S4A). Since the functional form in all cases is p active = (1 + w I/A ) − , where w I/A isthe total weight of the inactive states divided by the total weight of the active states in theappropriate limit (as seen in Figure 3A), a Boolean response (p active ≈ or ) can only beachieved when w I/A (cid:29) or w I/A (cid:28) , respectively. Hence, the values of w I/A at the eightdifferent limits of ligand concentration will determine the full logic response of the protein.Note that since cooperative interactions between ligands are absent in the MWC frame-work, the eight different w
I/A expressions depend on only four independent MWC param-eters, namely, { ∆ ε AI , γ , γ , γ } . Therefore, only four of the eight limiting w I/A values canbe independently tuned, and any w
I/A limit can be expressed as a function of four differ-ent and independent w
I/A limits, resulting in a constraint condition. Since each w
I/A is aproduct of some γ i ’s and e − β ∆ ε AI (Figure S4A), we look for constraint conditions that havea multiplicative form, namely, w s ∗ = (cid:89) i =1 w α n s n , (S43)where w s ∗ is the target limit, s n (cid:54) = s ∗ (1 ≤ n ≤ are the labels of four different limits and α n are real coefficients. Searching over all conditions of such form (see the supplementaryJupyter Notebook 2 for details), we identify a total of eight functionally unique constraints,w ij × w = w i × w j , (S44)w × w j = w ij × w jk , (S45)w ij × w k = w ik × w j , (S46)w × w = w ij × w k , (S47)w ij × w k = w × w ik × w jk , (S48)w × w = w × w × w , (S49)w × w = w × w × w , (S50)S10 × w i × w j = w ij × w k , (S51)where ≤ i, j, k ≤ .Further searching for a minimum set of constraints that can account for all gates in-compatible with the MWC framework, we identify the constraints in eqs S44-S47 as thenecessary and sufficient ones (see the supplementary Jupyter Notebook 2). Graphical repre-sentations of these four constraints on a cubic diagram are shown in Figure S4B. Note thatthese conditions are all of the form w s w s = w s w s , (S52)where s i are labels corresponding to different ligand concentration limits. Logic responseswhere w s , w s (cid:28) (cid:29) while w s , w s (cid:29) (cid:28) cannot be achieved, since they contradictthe constraint condition. Conditions 1 and 2 in Figure S4B, for example, demonstrate thatXOR and XNOR gates cannot be realized by any two ligands in the absence (condition 1) orpresence (condition 2) of a third ligand - a result expected from the 2-ligand analysis doneearlier. On the other hand, conditions 3 and 4 are specific to the 3-ligand response.Checking the 80 functionally unique gates against the four constraints in Figure S4B, weobtain a set of 34 functionally unique and MWC-compatible gates, 17 of which are shownin Figure S5A while the other half are their logical complements (i.e. ON ↔ OFF swappingis performed for each of the cube elements). (A) [L ]→0 [ L ] → [L ]→0 [ L ] → ∞ [L ]→∞ [L ]→∞ 1+γ e -βΔε AI w γ e -βΔε AI w γ γ e -βΔε AI w γ e -βΔε AI w [L ]→0 [ L ] → [ L ] → ∞ [L ]→∞ (B) j jiji i kk w
123 × w j = w ij × w jk i k0 k j j ij i k w
123 × w = w ij × w k e -βΔε AI w γ e -βΔε AI w -βΔε AI w e -βΔε AI w ij × w k = w jk × w i i k ji ij k k j w ij × w = w i × w j condition 1: condition 2: condition 3: condition 4: k j i k0 jiji k Figure S4. Three-ligand logic gates that are incompatible with the MWC framework. (A) Probability that the protein is active in the 8 different ligand concentration limits. The totalweight of the inactive states relative to the active states is indicated in gray for all limits. (B)Cubic diagrams of logic responses that are incompatible with the MWC framework, along with theconstraint equations used to obtain them. The limits relevant to the constraint conditions areshown in color, and a transparent gray plane containing these relevant limits is added for clarity.In all four diagrams ≤ i , j , k ≤ . S11 .2 Logic Switching
Here we describe how the table of all possible logic switches inducible by a third ligand(Figure 6D) can be obtained from the list of MWC-compatible 3-ligand gates (Figure S5),and also derive the parameter conditions for AND → OR and AND → YES logic switches.As illustrated in Figure 6C, logic switching can be achieved by increasing the concentra-tion of any of the three ligands. Following the same procedure, we iterate over the list of gatesshown in Figure S5A and for each of them identify the set of possible logic switches. Theset of all logic switches present in Figure S5A together constitute the entries of the table inFigure 6D. Note that if a gate is compatible with the MWC framework, then its logical com-plement is also compatible, and therefore, the possibility of switching between two gates,Gate 1 → Gate 2, implies the possibility of switching between their logical complements,NOT (Gate 1) → NOT (Gate 2).
NONE → NONE (→|→ | → )NONE → NOR (→)OR → NONE (→ ) OR → AND (→ ) ANDNi → ALL (→) OR → OR (→ ) OR → ALL (→|→ | → )YESi → ALL (→| → )OR → YESi (→ )YESi → ORNi ( → )ANDNi → ORNj (→| → )ANDNi → NOTi (→| → ) NONE → ORNi (→) YESi → YESi (→ | → ) YESi → OR (→ | → )AND → ALL (→)NONE → ALL (→)YESi → AND ( → )ANDNi → YESj (→ )YESi → NONE ( → ) NONE → NAND (→)YESi → ANDNj (→ | → )ANDNi → ANDNi (→ )NONE → NOTi (→)ANDNi → NONE (→ | → ) NONE → AND (→|→ | → ) AND → AND ( → ) NONE → OR (→) AND → OR (→|→ | → )AND → YESi (→ | → )NONE → YESi (→|→ )NONE → ANDNi (→|→ )AND → NONE ( → ) AND OR YES i YES j NONE NAND NOR ORN i NOT i ORN j NOT j ALLANDN i ANDN j L i L j (A)(B) Figure S5. Functionally unique 3-ligand MWC gates and possible schemes of logicswitching. (A) List of functionally unique 3-ligand MWC gates that have an inactive base state(in the absence of ligands). The set of logic switches that can be achieved by increasing theconcentration of one of the ligands is listed on the bottom of each gate, with the gray arrowsindicating the corresponding directions of increasing ligand concentration. Transitions withswapped labels (i ↔ j) are also possible and are not listed. Arrows corresponding to the ligandaxes on different faces of the cube are included to assist the derivation of possible logic switches.(B) Schematics of 2-ligand gates adapted from Figure 6D for convenience. S12ow, we show how an MWC protein can exhibit the switching behaviors in Figure 7B,D(AND → OR and AND → YES ) by saturating the concentration of the third ligand. We firstconsider the behavior of the protein in the absence of the third ligand ( [ L ] = 0 , with p active limits given in Figure S4A, left) and then consider how the protein acts at the saturatingconcentration of the third ligand ( [ L ] → ∞ , with p active limits given in Figure S4A, right).With [ L ] = 0 , the protein ignores the third ligand and behaves identically to a protein withN = 2 ligands. In the limit [ L ] → ∞ , however, the protein behaves as if it only has twoligands with an altered free energy difference ∆ ε (cid:48) AI between the active and inactive statesgiven by ∆ ε (cid:48) AI = ∆ ε AI − k B T log γ . (S53)Suppose that a protein acts as an AND gate when [ L ] = 0 and transitions into an ORgate when [ L ] → ∞ , as in Figure 7B. From Figure 3B, the MWC parameters must satisfy γ , γ (cid:28) e − β ∆ ε AI (cid:28) γ γ (S54)in the absence of L (AND behavior) and (cid:28) e − β ∆ ε (cid:48) AI (cid:28) γ , γ (S55)when [ L ] is saturating (OR behavior). Using eqs S53, we can rewrite the condition S55 as γ (cid:28) e − β ∆ ε AI (cid:28) γ γ , γ γ . (S56)Combining eq S54 and eq S56, we find the second condition reported in Figure 7A, namely, γ , γ , γ (cid:28) e − β ∆ ε AI (cid:28) γ γ , γ γ , γ γ . (S57)The first condition in Figure 7A is then obtained by using the outer inequalities, that is, γ k (cid:28) γ i γ j ⇒ γ i γ j (cid:28) γ k and (S58) γ i (cid:28) γ i γ k ⇒ γ k (cid:28) . (S59)Lastly, we derive the parameter conditions needed to achieve an AND → YES switchingby saturating the third ligand. Conditions for the AND behavior in the absence of the thirdligand are already known (eq S54). To achieve a YES gate, p active at [ L ] → ∞ needs tomeet the following limits: p , , ∞ = 11 + γ e − β ∆ ε AI ≈ , (S60)S13 , ∞ , ∞ = 11 + γ γ e − β ∆ ε AI ≈ , (S61)p ∞ , , ∞ = 11 + γ γ e − β ∆ ε AI ≈ , (S62)p ∞ , ∞ , ∞ = 11 + γ γ γ e − β ∆ ε AI ≈ . (S63)These limits suggest constraints on ∆ ε AI , which, combined with eq S54, result in γ , γ , γ , γ γ (cid:28) e − β ∆ ε AI (cid:28) γ γ , γ γ , γ γ , γ γ γ . (S64)The outer inequalities, in turn, suggest conditions for the γ parameters, namely, γ i (cid:28) γ i γ k ⇒ γ k (cid:28) , (S65) γ γ (cid:28) γ γ ⇒ γ (cid:28) γ , (S66) γ γ (cid:28) γ γ ⇒ γ (cid:28) γ . (S67)Accounting for these additional constraints, eq S64 simplifies into γ , γ γ (cid:28) e − β ∆ ε AI (cid:28) γ γ ,,
NONE → NONE (→|→ | → )NONE → NOR (→)OR → NONE (→ ) OR → AND (→ ) ANDNi → ALL (→) OR → OR (→ ) OR → ALL (→|→ | → )YESi → ALL (→| → )OR → YESi (→ )YESi → ORNi ( → )ANDNi → ORNj (→| → )ANDNi → NOTi (→| → ) NONE → ORNi (→) YESi → YESi (→ | → ) YESi → OR (→ | → )AND → ALL (→)NONE → ALL (→)YESi → AND ( → )ANDNi → YESj (→ )YESi → NONE ( → ) NONE → NAND (→)YESi → ANDNj (→ | → )ANDNi → ANDNi (→ )NONE → NOTi (→)ANDNi → NONE (→ | → ) NONE → AND (→|→ | → ) AND → AND ( → ) NONE → OR (→) AND → OR (→|→ | → )AND → YESi (→ | → )NONE → YESi (→|→ )NONE → ANDNi (→|→ )AND → NONE ( → ) AND OR YES i YES j NONE NAND NOR ORN i NOT i ORN j NOT j ALLANDN i ANDN j L i L j (A)(B) Figure S5. Functionally unique 3-ligand MWC gates and possible schemes of logicswitching. (A) List of functionally unique 3-ligand MWC gates that have an inactive base state(in the absence of ligands). The set of logic switches that can be achieved by increasing theconcentration of one of the ligands is listed on the bottom of each gate, with the gray arrowsindicating the corresponding directions of increasing ligand concentration. Transitions withswapped labels (i ↔ j) are also possible and are not listed. Arrows corresponding to the ligandaxes on different faces of the cube are included to assist the derivation of possible logic switches.(B) Schematics of 2-ligand gates adapted from Figure 6D for convenience. S12ow, we show how an MWC protein can exhibit the switching behaviors in Figure 7B,D(AND → OR and AND → YES ) by saturating the concentration of the third ligand. We firstconsider the behavior of the protein in the absence of the third ligand ( [ L ] = 0 , with p active limits given in Figure S4A, left) and then consider how the protein acts at the saturatingconcentration of the third ligand ( [ L ] → ∞ , with p active limits given in Figure S4A, right).With [ L ] = 0 , the protein ignores the third ligand and behaves identically to a protein withN = 2 ligands. In the limit [ L ] → ∞ , however, the protein behaves as if it only has twoligands with an altered free energy difference ∆ ε (cid:48) AI between the active and inactive statesgiven by ∆ ε (cid:48) AI = ∆ ε AI − k B T log γ . (S53)Suppose that a protein acts as an AND gate when [ L ] = 0 and transitions into an ORgate when [ L ] → ∞ , as in Figure 7B. From Figure 3B, the MWC parameters must satisfy γ , γ (cid:28) e − β ∆ ε AI (cid:28) γ γ (S54)in the absence of L (AND behavior) and (cid:28) e − β ∆ ε (cid:48) AI (cid:28) γ , γ (S55)when [ L ] is saturating (OR behavior). Using eqs S53, we can rewrite the condition S55 as γ (cid:28) e − β ∆ ε AI (cid:28) γ γ , γ γ . (S56)Combining eq S54 and eq S56, we find the second condition reported in Figure 7A, namely, γ , γ , γ (cid:28) e − β ∆ ε AI (cid:28) γ γ , γ γ , γ γ . (S57)The first condition in Figure 7A is then obtained by using the outer inequalities, that is, γ k (cid:28) γ i γ j ⇒ γ i γ j (cid:28) γ k and (S58) γ i (cid:28) γ i γ k ⇒ γ k (cid:28) . (S59)Lastly, we derive the parameter conditions needed to achieve an AND → YES switchingby saturating the third ligand. Conditions for the AND behavior in the absence of the thirdligand are already known (eq S54). To achieve a YES gate, p active at [ L ] → ∞ needs tomeet the following limits: p , , ∞ = 11 + γ e − β ∆ ε AI ≈ , (S60)S13 , ∞ , ∞ = 11 + γ γ e − β ∆ ε AI ≈ , (S61)p ∞ , , ∞ = 11 + γ γ e − β ∆ ε AI ≈ , (S62)p ∞ , ∞ , ∞ = 11 + γ γ γ e − β ∆ ε AI ≈ . (S63)These limits suggest constraints on ∆ ε AI , which, combined with eq S54, result in γ , γ , γ , γ γ (cid:28) e − β ∆ ε AI (cid:28) γ γ , γ γ , γ γ , γ γ γ . (S64)The outer inequalities, in turn, suggest conditions for the γ parameters, namely, γ i (cid:28) γ i γ k ⇒ γ k (cid:28) , (S65) γ γ (cid:28) γ γ ⇒ γ (cid:28) γ , (S66) γ γ (cid:28) γ γ ⇒ γ (cid:28) γ . (S67)Accounting for these additional constraints, eq S64 simplifies into γ , γ γ (cid:28) e − β ∆ ε AI (cid:28) γ γ ,, γ γ ,,